Correlations of almost primes. (9th March 2023)

• Record Type: Journal Article
• Authors: EVANS, NATALIE
• Is Part Of: Mathematical proceedings of the Cambridge Philosophical Society. Volume 174:Part 2(2023)
• Page Start: 301
• Page End: 344
• Publication Date: 2023-03-09
• Subjects: 11N37; Mathematics -- Periodicals; 510.5
• DOI: 10.1017/S0305004122000251
• Languages: English
• ISSNs: 0305-0041

Abstract

Abstract: We prove that analogues of the Hardy–Littlewood generalised twin prime conjecture for almost primes hold on average. Our main theorem establishes an asymptotic formula for the number of integers $n=p_1p_2 \leq X$ such that $n+h$ is a product of exactly two primes which holds for almost all $|h|\leq H$ with $\log^{19+\varepsilon}X\leq H\leq X^{1-\varepsilon}$, under a restriction on the size of one of the prime factors of n and $n+h$ . Additionally, we consider correlations $n, n+h$ where n is a prime and $n+h$ has exactly two prime factors, establishing an asymptotic formula which holds for almost all $|h| \leq H$ with $X^{1/6+\varepsilon}\leq H\leq X^{1-\varepsilon}$ .